Ultimixt-package: set of R functions for estimating the parameters of mixture...

In Ultimixt: Bayesian Analysis of Location-Scale Mixture Models using a Weakly Informative Prior

Description

Despite a comprehensive literature on estimating mixtures of Gaussian distributions, there does not exist a well-accepted reference Bayesian approach to such models. One reason for the difficulty is the general prohibition against using improper priors (Fruhwirth-Schnatter, 2006) due to the ill-posed nature of such statistical objects. Kamary, Lee and Robert (2017) took advantage of a mean-variance reparametrisation of a Gaussian mixture model to propose improper but valid reference priors in this setting. This R package implements the proposal and computes posterior estimates of the parameters of a Gaussian mixture distribution. The approach applies with an arbitrary number of components. The Ultimixt R package contains an MCMC algorithm function and further functions for summarizing and plotting posterior estimates of the model parameters for any number of components.

Details

Package:	Ultimixt
Type:	Package
Version:	2.1
Date:	2017-03-07
License:	GPL (>=2.0)

Beyond simulating MCMC samples from the posterior distribution of the Gaussian mixture model, this package also produces summaries of the MCMC outputs through numerical and graphical methods.

Note: The proposed parameterisation of the Gaussian mixture distribution is given by

f(x| μ, σ , {\bf p}, \varphi, {\bf \varpi, ξ})=∑_{i=1}^k p_i f≤ft(x| μ + σ γ_i/√{p_i}, σ η_i/√{p_i}\right)

under the non-informative prior π(μ, σ)=1/σ. Here, the vector of the γ_i=\varphi Ψ_i\Big({\bf \varpi}, {\bf p}\Big)_i's belongs to an hypersphere of radius \varphi intersecting with an hyperplane. It is thus expressed in terms of spherical coordinates within that hyperplane that depend on k-2 angular coordinates \varpi_i. Similarly, the vector of η_i=√{1-\varphi^2}Ψ_i\Big({\bf ξ}\Big)_i's can be turned into a spherical coordinate in a k-dimensional Euclidean space, involving a radial coordinate √{1-\varphi^2} and k-1 angular coordinates ξ_i. A natural prior for \varpi is made of uniforms, \varpi_1, …, \varpi_{k-3}\sim U[0, π] and \varpi_{k-2} \sim U[0, 2π], and for \varphi, we consider a beta prior Beta(α, α). A reference prior on the angles ξ is (ξ_1, …, ξ_{k-1})\sim U[0, π/2]^{k-1} and a Dirichlet prior Dir(α_0, …, α_0) is assigned to the weights p_1, …, p_k.

For a Poisson mixture, we consider

f(x|λ_1, …, λ_k)=\frac{1}{x!}∑_{i=1}^k p_i λ_i^x e^{-λ_i}

with a reparameterisation as λ=\bf{E}[X] and λ_i=λ γ_i/p_i. In this case, we can use the equivalent to the Jeffreys prior for the Poisson distribution, namely, π(λ)=1/λ, since it leads to a well-defined posterior with a single positive observation.

Author(s)

Kaniav Kamary

Maintainer: [e-mail]

References

Fruhwirth-Schnatter, S. (2006). Finite Mixture and Markov Switching Models. Springer-Verlag, New York, New York.

Kamary, K., Lee, J.Y., and Robert, C.P. (2017) Weakly informative reparameterisation for location-scale mixtures. arXiv.

See Also

Ultimixt

Examples

	#K.MixReparametrized(faithful[,2], k=2, alpha0=.5, alpha=.5, Nsim=10000)

Ultimixt documentation built on May 1, 2019, 10:56 p.m.